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%I #20 Jul 19 2024 14:27:56
%S 144,192,216,240,288,336,360,384,432,480,504,540,576,600,648,672,720,
%T 768,792,840,864,900,936,960,1008,1056,1080,1152,1200,1248,1260,1296,
%U 1320,1344,1440,1512,1536,1560,1584,1620,1632,1680,1728,1800,1824,1848,1872,1920,1944,1980,2016,2040,2100,2112,2160,2240
%N Numbers k such that A245212(k) < 0.
%C If d are divisors of k then values of sequence A245212(k) are by bending moments in point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length k with bracket in point 0 by the schema: A245212(k) = (k * tau(k)) - Sum_{(d<k) | k} (d * tau(d)).
%C Numbers k such that A038040(k) = k * tau(k) < A245211(k) = Sum_{(d<k) | k} (d * tau(d)).
%C From _Amiram Eldar_, Jul 19 2024: (Start)
%C Numbers whose divisors have a mean abundancy index that is larger than 2.
%C The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 24, 243, 2571, 25583, 254794, 2551559, 25514104, 255112225, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02551... .
%C The least odd term in this sequence is a(276918705) = 10854718875. (End)
%H Jens Kruse Andersen, <a href="/A245214/b245214.txt">Table of n, a(n) for n = 1..10000</a>
%e Number 144 is in sequence because 144 * tau(144) = 2160 < Sum_{(d<144) | 144} (d * tau(d)) = 2226.
%t f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2500], s[#] > 2 &] (* _Amiram Eldar_, Jul 19 2024 *)
%o (Magma) [n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt 0]
%o (PARI) isok(n) = (n*numdiv(n) - sumdiv(n, d, (d<n)*d*numdiv(d))) < 0; \\ _Michel Marcus_, Aug 06 2014
%o (PARI) is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i,1]; e=f[i,2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2;} \\ _Amiram Eldar_, Jul 19 2024
%Y Cf. A000005, A038040, A245211, A245212, A245213.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Jul 23 2014